Linearized impulse wave propagating down a vertical column of heavy particles

A granular column is subjected to a small amplitude impact on its top. For a generalized power-law contact force between neighboring grains, numerical simulations show that the propagation of the impulse wave is controlled by dispersion. This leads quantitatively to a power-law decrease of the amplitude of the wave with depth. We find numerically the dependence of this power-law exponent on the force-law exponent. An analytic expression for the decrease is then derived from a long-wave approximation.